TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ACTUAL-ONEFORM-COUPLED-SOURCE-RESPONSE.txt
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000001 | THE ORIGINAL INNER-FLUCTUATION FAMILY HAS A FULL COUPLED TWO-FACE RESPONSE
000002 | 30 October 2026. Source-bound analytical continuation and direct v252 control.
000003 | Verdict: ESTABLISHED_SCOPED. No original, ledger or Atlas change.
000004 | 
000005 | 1. Actual family; distinguish reuse from the new calculation
000006 | Use the actual full96 coefficient of verification/v252_full_finite_triple.py:
000007 | full_D:102-116, grading:119-121, rep:139-168, control inputs:202-212.
000008 | The inner-one-form prescription is printed in v254_inner_fluctuations.py:6-12.
000009 | The explicit Higgs witness is already proved in original_insertions/PROOF.txt:
000010 | 29-49 and seam-joint-source-origin-audit-20261016/HIGGS_VARIATION_PROOF.txt:
000011 | 135-182. Its existence is reused, not a new origin result.
000012 | 
000013 | Set a=pi(0,I2,0), b=pi(1,0,0), A=a[D_F,b]. The actual real selfadjoint
000014 | completion A+A*+J_F(A+A*)J_F^-1 is D_Y=full_D(Yu,Yd,Ye,Ynu,0).
000015 | Consequently the existing real radial Higgs direction gives
000016 |   Phi_s=D_F+s D_Y=(1+s)D_Y+D_MR.                         (1)
000017 | All four Yukawa matrices are still the original supplied coefficient inputs;
000018 | the Majorana block is retained unchanged. No new field family is introduced.
000019 | 
000020 | The NEW calculation is the response of BOTH faces and their crossing block
000021 | along (1), at the already coupled background D_F, including its Majorana
000022 | coupling. It does not replace D_F by zero to obtain a commuting derivative.
000023 | 
000024 | 2. Whole source, actual domains and noncommutative derivative
000025 | Reuse graded_boundary_functor/FINITE-COUPLED.txt:10-68 and the existing
000026 | pfaff_source_review/NORMAL-KINETIC-PAIRING.txt:sections1-4. In the constant
000027 | common-quarter product chart, p=n+alpha, alpha=1/4 or3/4, let ell>0 and put
000028 |   h_s=p gamma_F-i gamma_F Phi_s,
000029 |   V=-i gamma_F D_Y, T_s=p^2+Phi_s^2,
000030 |   h_s^2=T_s, dot(T)_0={D_F,D_Y}=:Q.
000031 | The finite coefficient is bounded. Thus the original H1 operator domain is
000032 | unchanged, and both total Clifford grades and the original adjoint slots
000033 | remain in the full T_faces chart. No finite carrier or chirality is removed.
000034 | 
000035 | The given charged two-face covariance is the ENTIRE projector
000036 |   G_s=[[a_s,X_s],[X_s,1-a_s]],
000037 |   a_s=(I+tanh(ell h_s))/2, X_s=(2cosh(ell h_s))^-1.
000038 | The conjugate block is I-bar(G_s), as already proved for T_faces.
000039 | Define, for u>0,
000040 |   q(u)=tanh(ell sqrt(u))/sqrt(u),
000041 |   x(u)=1/(2cosh(ell sqrt(u))).
000042 | At s=0 the exact whole response is
000043 |   dot(a)=1/2 [V q(T)+h Dq_T(Q)],
000044 |   dot(X)=Dx_T(Q),
000045 |   dot(G)=[[dot(a),dot(X)],[dot(X),-dot(a)]].              (2)
000046 | Here h=h_0 and T=T_0. In an eigenbasis of T,
000047 |   (Df_T(Q))_ij=f[1](t_i,t_j) Q_ij,
000048 | where f[1](u,v)=(f(u)-f(v))/(u-v) and f[1](u,u)=f'(u).
000049 | It is generally WRONG to replace Df_T(Q) by f'(T)Q.
000050 | 
000051 | The first diagonal response is nonzero whenever D_Y is nonzero: equivalently
000052 | dot(a)=D(tanh(ell h))[V]/2. Every divided difference of tanh(ell u) on the
000053 | real spectrum, including its diagonal derivative, is strictly positive for
000054 | finite u and ell>0. This finite-matrix Frechet map is injective. This proves
000055 | nonvanishing without assuming [T,Q]=0 or selecting a favorable matrix entry.
000056 | 
000057 | The need for the noncommutative derivative is already visible on the actual
000058 | one-generation neutrino path: DY has edges01,23=m, DMR has edge13=M and
000059 | gamma=diag(1,-1,-1,1). For real m,M,
000060 |   [D_F^2,{D_F,D_Y}]=M^3 m
000061 |        [[0,0,0,-1],[0,0,1,0],[0,-1,0,0],[1,0,0,0]],
000062 | whose Hilbert-Schmidt norm squared is 4 M^6 m^2. The same holds with the
000063 | scalar p^2 added. This is an actual block witness, not a different source.
000064 | 
000065 | 3. All modes and all already given field products
000066 | For fixed finite D_F,D_Y and large |p|, dot(a)=O(1/|p|) while
000067 | dot(X)=O(exp(-ell|p|)/|p|). Therefore the complete circle response dot(G)
000068 | is Hilbert-Schmidt; its crossing derivative is trace class. No cutoff limit
000069 | is used as a substitute for this all-mode estimate.
000070 | At every s the same spectral-calculus G_s is an orthogonal projector, hence
000071 |   G dot(G)+dot(G) G=dot(G).
000072 | The existing pure-projector CAR theorem fixes all words of the same given
000073 | source. Their derivatives follow from (2) by multilinearity. For example,
000074 | if C_ab(s)=omega_s(a*(u_a)a(v_b)), the normally ordered n-pair expectation
000075 | is det(C(s)); its derivative is the sum of determinants with one row
000076 | replaced by the derivative row. Thus (2) supplies the joint response of the
000077 | whole existing field source, rather than unrelated responses on each face.
000078 | This is a background-dependent source response, not a bosonic field average.
000079 | 
000080 | 4. Direct actual-constructor control already executed
000081 | The saved computation imports the original v252 builder without calling
000082 | its run(). It reproduces exactly that module's seed252 input construction,
000083 | and uses p=1/4, ell=1. These are the existing product-chart mode and a unit
000084 | normal-width arithmetic control, not a physical-time or mass prediction.
000085 |   ||[D_F,D_Y]||_HS =21.72288631787473
000086 |   ||dot(a)||_HS    =1.26406156359676
000087 |   ||dot(X)||_HS    =1.9546014016433078
000088 |   ||dot(G)||_HS    =3.291904699674238
000089 | The inner-form identity, oddness, Hermiticity, J_F reality, h^2=T and
000090 | delta(h^2)=Q have zero numerical residual; unchanged Majorana residual is
000091 | 1.86e-15. G^2=G residual is1.19e-14; projector-tangent residual1.25e-14.
000092 | The central-difference comparison of the ENTIRE G has residual9.96e-10
000093 | at step1e-5. This is one focused builder control, not a suite or promotion.
000094 | Data: workspace outputs/multi-model-source-search-20261029/
000095 | ACTUAL-INNER-FLUCTUATION-JOINT-BOUNDARY-RESPONSE.json
000096 | SHA256 1c7d4d5a985f64d22eb008c9bb01663a8bcd8ffdb4a66480b1963c68b99399b5.
000097 | 
000098 | 5. What the original joint-action passages actually assert
000099 | The corpus DOES print fermionic terms. The older finite operator basis in
000100 | documents/archive--paper-latex--tfpt-32/sections/
000101 | 013-subsection-rg-map-for-compiler-outputs.tex:184-201 includes both
000102 | bar(psi) i slash(D) psi and bar(psi) Phi psi explicitly at194-195.
000103 | That is an RG operator-basis statement, not an explicit joint measure.
000104 | 
000105 | The stronger jointly printed action is at documents/tfpt_research_contracts/
000106 | sections/017-source.tex:153-191, equation161-164:
000107 |   Z_a[J]=int DU Dpsi DPhi exp(-S_a+J O),
000108 |   S_a=S_Wilson[U]+bar(psi)D_GW[U,Phi]psi+S_Phi[U,Phi]+S_seam/top.
000109 | Its exact status is Research Contract TFPT4D.LATTICE.ACTION.01, OPEN:
000110 | the text says "Demand. Exhibit one fully explicit finite family" at159.
000111 | The later positive Hamiltonian, coupled link-wall and dynamics-class updates
000112 | at193-235 are retained; their text explicitly does not promote this contract.
000113 | They must not be erased by quoting only the earlier sentence at190.
000114 | The bosonic spectral expansion v255 is not, by itself, this joint fermionic
000115 | action. No requirement S_bos=-logPf or bosonic Hessian=Dirac is imposed here.
000116 | 
000117 | The existing normal L=-ND=partial_r+B presentation supplies the charged
000118 | cross pairing and its regulated Pfaffian=det(K), with its already stated
000119 | boundary/Berezin-orientation conditions. It is a kinetic presentation of
000120 | the given graph source. Its determinant scalar has not thereby been equated
000121 | to the original unnormalized S_prim/Z_adm source. A complete operational Seed
000122 | already contains its state and time; no second state is requested for it.
000123 | 
000124 | 6. Preserve the original bound reference, gap and reflection types
000125 | In the original primitive core, D_ref and D_rel are the even and odd parts
000126 | under the ACTUAL geometric reflection/deck operation. They are not freely
000127 | independent backgrounds. Their original binding must remain in any later
000128 | variation or joint source call. This note does not substitute G's involution
000129 | for that operation or identify the given graph with the full primitive measure.
000130 | The Cauchy polarization involution iota_C=2G-I has dot(iota_C)=2dot(G).
000131 | It is distinct from geometric R, OS Theta and Q_Theta.
000132 | 
000133 | Original chi_seed is the inverse positive spectral gap of |B_Sigma|:
000134 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/
000135 | 001-section-primitive-core-and-carrier-interface-handoff.tex:42-55.
000136 | The numerical gap jet in the JSON is only the jet of THIS product-chart h.
000137 | It computes chi_seed only if the original B_Sigma is actually intertwined
000138 | with the stated full boundary operator. No abstract P1 gap identity is inferred.
000139 | 
000140 | The gained arrow is the whole, noncommuting, already coupled two-face source
000141 | response along an actual original inner-fluctuation direction. Its coupling
000142 | values remain the supplied original D_F inputs. Their original selection and
000143 | the full bosonic background integration are not supplied by this derivative.
000144 | No new freely selected source, measure, time, branch or regulator was added.